“Average Dice” part 2 — Damage Rolls

In the last installment of this series, I looked at the math behind attack rolls in Warmachine.  This time, I’m going to move onto damage rolls, as once you scratch the surface and get past the notion that “average dice equals seven,” there is some interesting stuff going on in the probabilities.

seven1.jpg

Fascinating.  Tell me more…

For the uninitiated, Warmachine uses 2d6 to resolve most attack and damage rolls.  Generally, when making an attack against an enemy model, you must roll 2d6, add your attack modifier, and equal or beat the opposing model’s DEF to score a hit.  Once you have hit, you determine how much damage you do by again rolling 2d6, adding the POW/P+S of the weapon you are attacking with, and comparing it to the ARM value of the opposing model.  For every point that this roll exceeds ARM, you do one point of damage to the opposing model.  Models in Warmachine have anywhere from one to sixty or more hit points or “boxes,” and to make it easier, players usually mentally subtract the opposing player’s ARM from their POW before rolling damage — calculating that, for example, with a POW of 16 up against an ARM of 20, they will do 2d6-4 or “dice minus four” damage.

Single wound or multi-wound?

Before I get started, I would like to point out that there is a difference between single wound and multi-wound models.  For single wound models, you only have to do a single point of damage to kill them, which means that you get the same result (usually, a dead enemy model) whether your roll beats their ARM by one or by a lot.  Since we don’t care how much we overkill a single-wound model by**, our math actually resembles the to-hit rolls we discussed last week in that we’re essentially rolling to hit a target number, which in this case is one higher than their ARM value, in order to do at least one point of damage and kill the opposing model.

Additionally, for a multi-wound model, you can also use a similar principle to calculate your odds of one-shot killing the model, by simply adding up the number of boxes and the ARM to figure out what you need to roll to one-shot kill the enemy model.

Since the math on trying to roll a target number or better and not caring how much you beat the target number by was discussed extensively last week, right now I’m going to focus on a case where you are up against a multi-wound model, and assume you are just trying to do as much damage as you can.

Widowmakers vs. Forge Seer

A couple weeks ago, I was playing a game and having some Khador-on-Khador action. My opponent had a Greylord Forge Seer marshalling a warjack, and had left it in the open and within striking distance of an entire four-man squad of Widowmaker Scouts and a single Widowmaker Marksman.

F_Khador_WidowmakerDue to the accuracy of the sniper rifles on the Widowmakers (who says the Khadorans don’t have a knack for precision engineering?), hitting the target wasn’t a big problem.  At anything but snakes to hit, each individual shot had a 97.2% chance of connecting and I have an over 85% chance of hitting with all five.

The problem, however, comes when it is time to do damage.  The opposing Greylord Forge Seer has an ARM of 16 and eight boxes.  On the Widowmakers, I have a choice.  Due to their Sniper rule, with the Scouts, on a hit, I can choose to either do one point of damage automatically or roll a POW 10, and with the Marksman, I can choose to either do three points of damage or roll a POW 12.

Adding up the automatic damage, we see that if I take advantage of the Sniper rule, doing one point of damage four times and three points of damage once, I will end up doing seven points of damage — just shy of what I need to kill this Forge Seer.  Since I would rather him not survive until the next turn, that won’t do.

So, if I want any chance of killing him at all, I use my POW 10s and POW 12s and roll it out.  Here is where the math gets interesting…

Dice minus X…

“Average dice” theory states that since “average dice” on 2d6 equals seven, my Scouts, doing 2d6-6 damage, will on average do one point of damage each, while my Marksman, doing 2d6-4, will on average do three points.  Again, this adds up to seven damage, which means that I will probably not manage to kill the Greylord Forge Seer.

Fortunately for me, the math of the “average dice” theory is wrong.  To calculate the expected value of a random discrete variable such as a series of die rolls, it’s a simple matter of, for each possible outcome, multiplying the probability by the outcome.  The formula for the expected value*** on a damage roll, thus, is:

E(x) = x1p1 + x2p2 + x3p3 + … xnpn

Where, for each possible outcome on the dice, p is the probability of said outcome and x is the damage.

When we start calculating this for 2d6, we start to notice something interesting…

expected_value_damage.png

Here is the calculations, with the main part of the table showing the value of each term in the equation for all cases from dice-2 to dice-11, with the EV in the bottom row

From dice minus two upwards, for every +1 you add to the roll, the expected value of your damage increases by one.  In this area, the “average dice” theory seems to more or less work out, in that the expected value of 2d6-1 is equal to 6, the expected value of 2d6 is equal to 7, and so on.  However, as we go into dice minus three and lower, the curve starts to flatten out and trail off, only hitting zero when we get to dice minus 12 and can’t possibly roll a thirteen or better on two dice.

expected_value_damage_graph.png

The blue line is correct math; the orange line is wrong

This is because you can’t roll negative damage.  Think of it this way — if you are rolling damage at dice minus seven, and you roll snakes (2) once and boxcars (12) once, you have rolled perfectly average over those two rolls, but you still did five points of damage.  That time you rolled snakes, you didn’t heal the opposing warjack for five points of damage, you simply failed to crack ARM and did nothing.  As such, you did five more points of damage than the “average dice equals seven” theory would imply because your low roll didn’t “cancel out” your high roll in any way.

Looking at the blue curve, even at dice minus seven and below, you still have a chance of rolling high and doing damage.  In fact, the expected value on 2d6-7 damage is almost a full point higher than what the “average dice” theory states!

So, what does this mean?

Remember my situation with the Widowmakers and the Forge Seers?  “Average Dice” implies that I would do seven damage, but if we add up the expected values of four shots at 2d6-6 and one shot at 2d6-4, we get an expected value of 9.333.  With the Forge Seer having eight boxes, that means that assuming I hit all my shots, I have a pretty good chance of killing it.

Which is what I did — failing to crack armour on the first two shots, rolling boxcars on the third for six damage, using the fourth to kill some random single-wound mook, and then using the Marksman to do an automatic three points to finish him off.

To-hit and damage

Of course, this is Warmachine and you can always miss.  If you want to factor in your odds of hitting, all you have to do is multiply the expected value on your damage roll by the probability that your attack roll will hit.  In this case, since I need not snakes on all my shots, I simply multiply the EV by my the probaility of hitting.  In the Widowmakers case, it’s an easy calculation, multiplying the EV by 0.972, as I have a 97.2% chance of not rolling snakes, but this can be done for any to-hit value and dice plus/minus value on damage

EV of damage.png

Memorize this chart to be good at Warmachine…

Final thoughts

  • Against single-wound models, or when you want to calculate the probaility that you will one-shot something, you can simply adapt last week’s to-hit results
  • Below 2d6-2, the “average dice” theory of calculating damage breaks down
  • The expected value on your damage roll at this point is higher than you might otherwise expect, because low rolls do zero damage instead of negative damage that might cancel out the high rolls
  • To factor in your chance of missing, simply multiply the EV of your damage roll by your hit probability, and you can figure out the EV of the amount of damage you will do before you even make the attack roll.
  • For multiple attacks, simply add up the EV of each individual attack, factoring in your hit probability.  Doing so, you can figure out the “DPS” of a multi-attacking model such as a warjack or warbeast against models of a certain DEF and ARM value.
  • As always, the dice gods are fickle and literally any possible outcome, even some extreme outcome with very low odds, can happen.

Stay tuned next time for when I discuss what happens when you start throwing additional dice into the mix, and maybe if you’re lucky, why Vlad1 is really really good!

 

**Except maybe for bragging rights and to laugh at how badly some guy got splattered.  Like last weekend, where I overkilled a Cygnaran warcaster by 34 points…

***Again, I’m using the term “expected value” in the mathematical sense to represent the mean outcome, weighted by probability.  Over hundreds and thousands of rolls, you would expect your average outcome to converge on this value.  But on one individual trial, you can still roll anything from snakes to boxcars.  Just because something is the expected value, don’t count on it, because about 50% of the time, you will be disappointed.

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